cohen-macaulay $r$-partite graphs with minimal clique cover

Cohen-Macaulay $r$-partite graphs with minimal clique cover

‎In this paper‎, ‎we give some necessary conditions for an $r$-partite graph such that the edge ring of the graph is Cohen-Macaulay‎. ‎It is proved that if there exists a cover of an $r$-partite Cohen-Macaulay graph by disjoint cliques of size $r$‎, ‎then such a cover is unique‎.

Bi-Cohen-Macaulay graphs

In this paper we consider bi-Cohen-Macaulay graphs, and give a complete classification of such graphs in the case they are bipartite or chordal. General biCohen-Macaulay graphs are classified up to separation. The inseparable bi-CohenMacaulay graphs are determined. We establish a bijection between the set of all trees and the set of inseparable bi-Cohen-Macaulay graphs.

Unmixed $r$-partite graphs

‎Unmixed bipartite graphs have been characterized by Ravindra and‎ ‎Villarreal independently‎. ‎Our aim in this paper is to‎ ‎characterize unmixed $r$-partite graphs under a certain condition‎, ‎which is a generalization of Villarreal's theorem on bipartite‎ ‎graphs‎. ‎Also, we give some examples and counterexamples in relevance to this subject‎.

Sequentially Cohen-Macaulay Graphs of Form θ_{n1, ...nk}

Whiskers and sequentially Cohen-Macaulay graphs

Let G be a simple (i.e., no loops and no multiple edges) graph. We investigate the question of how to modify G combinatorially to obtain a sequentially CohenMacaulay graph. We focus on modifications given by adding configurations of whiskers to G, where to add a whisker one adds a new vertex and an edge connecting this vertex to an existing vertex in G. We give various sufficient conditions and...

A New Construction for Cohen-macaulay Graphs

Let G be a finite simple graph on a vertex set V (G) = {x11, . . . , xn1}. Also let m1, . . . , ,mn ≥ 2 be integers and G1, . . . , Gn be connected simple graphs on the vertex sets V (Gi) = {xi1, . . . , ximi}. In this paper, we provide necessary and sufficient conditions on G1, . . . , Gn for which the graph obtained by attaching the Gi to G is unmixed or vertex decomposable. Then we character...